2.2 Heat
51
2.6 a. Obtain a formula for the work done in reversibly and
isothermally compressing 1.000 mol of a gas from a
volume V 1 to a volume V 2 if the gas obeys the
Redlich–Kwong equation of state.
b. Using the formula from part a, find the work done in
reversibly compressing 1.000 mol of carbon dioxide
from 10.00 L to 5.000 L at 298.15 K. Compare with the
result obtained by assuming that the gas is ideal.
c. Using the formula from part a, calculate the work done
on the surroundings if 1.000 mol of carbon dioxide
expands isothermally but irreversibly from 5.000 L to
10.00 L at an external pressure of 1.000 atm. Compare
with the result obtained by assuming that the gas is ideal.
2.7 a. Obtain a formula for the work done in reversibly and
isothermally compressing 1.000 mol of a van der Waals
gas from a volume V 1 to a volume V 2 .
b. Using the formula from part a, find the work done in
reversibly compressing 1.000 mol of carbon dioxide
from 10.00 L to 5.000 L at 298.15 K. Compare with the
result obtained by assuming that the gas is ideal.
c. Using the formula from part a, calculate the work done
on the surroundings if 1.000 mol of carbon dioxide
expands isothermally but irreversibly from 5.000 L to
10.00 L at an external pressure of 1.000 atm. Compare
with the result obtained by assuming that the gas is ideal.
2.8 Test the following differentials for exactness:
a. T 2 dT + TVdV
b. xe xy dx + ye xy dy
c. xydx +
1
2 x 2 dy
2.9 Carry out the line integral:
c
nR
V
dT −
nRT
V 2 dV
on the path from (300.0 K, 10.0 L) to (300.0 K, 20.0 L) to
(400.0 K, 20.0 L). Take n 1.000 mol.
2.2
Heat
Joseph Black was the first to distinguish between the quantity of heat and the
“intensity” of heat (temperature) and to recognize latent heat absorbed or given
off in phase transitions. However, Black believed in the caloric theory of heat,
which incorrectly asserted that heat was an “imponderable” fluid called “caloric.”
This incorrect theory was not fully discredited until several decades after Black’s
death.
Joseph Black, 1728–1799, was a
Scottish chemist who discovered
carbon dioxide (“fixed air”) by heating
calcium carbonate.
Heat Transferred during Temperature Changes
A small amount of heat added to a system is proportional to its change in temperature
of the system if there is no phase change or chemical reaction:
dq CdT
(2.2-1)
where dq is an infinitesimal amount of heat transferred to the object and dT is a resulting
infinitesimal change in temperature. The proportionality constant C is called the heat
capacity of the system.
Equation (2.2-1) does not indicate that C is a derivative of q with respect to T . We
will see that dq is an inexact differential so that the heat capacity C depends on the way
in which the temperature of the system is changed. If the temperature is changed at
constant pressure, the heat capacity is denoted by C P and is called the heat capacity at
constant pressure. If the temperature is changed at constant volume, the heat capacity
is denoted by C V and is called the heat capacity at constant volume. These two heat
capacities are not generally equal to each other.
51
2.6 a. Obtain a formula for the work done in reversibly and
isothermally compressing 1.000 mol of a gas from a
volume V 1 to a volume V 2 if the gas obeys the
Redlich–Kwong equation of state.
b. Using the formula from part a, find the work done in
reversibly compressing 1.000 mol of carbon dioxide
from 10.00 L to 5.000 L at 298.15 K. Compare with the
result obtained by assuming that the gas is ideal.
c. Using the formula from part a, calculate the work done
on the surroundings if 1.000 mol of carbon dioxide
expands isothermally but irreversibly from 5.000 L to
10.00 L at an external pressure of 1.000 atm. Compare
with the result obtained by assuming that the gas is ideal.
2.7 a. Obtain a formula for the work done in reversibly and
isothermally compressing 1.000 mol of a van der Waals
gas from a volume V 1 to a volume V 2 .
b. Using the formula from part a, find the work done in
reversibly compressing 1.000 mol of carbon dioxide
from 10.00 L to 5.000 L at 298.15 K. Compare with the
result obtained by assuming that the gas is ideal.
c. Using the formula from part a, calculate the work done
on the surroundings if 1.000 mol of carbon dioxide
expands isothermally but irreversibly from 5.000 L to
10.00 L at an external pressure of 1.000 atm. Compare
with the result obtained by assuming that the gas is ideal.
2.8 Test the following differentials for exactness:
a. T 2 dT + TVdV
b. xe xy dx + ye xy dy
c. xydx +
1
2 x 2 dy
2.9 Carry out the line integral:
c
nR
V
dT −
nRT
V 2 dV
on the path from (300.0 K, 10.0 L) to (300.0 K, 20.0 L) to
(400.0 K, 20.0 L). Take n 1.000 mol.
2.2
Heat
Joseph Black was the first to distinguish between the quantity of heat and the
“intensity” of heat (temperature) and to recognize latent heat absorbed or given
off in phase transitions. However, Black believed in the caloric theory of heat,
which incorrectly asserted that heat was an “imponderable” fluid called “caloric.”
This incorrect theory was not fully discredited until several decades after Black’s
death.
Joseph Black, 1728–1799, was a
Scottish chemist who discovered
carbon dioxide (“fixed air”) by heating
calcium carbonate.
Heat Transferred during Temperature Changes
A small amount of heat added to a system is proportional to its change in temperature
of the system if there is no phase change or chemical reaction:
dq CdT
(2.2-1)
where dq is an infinitesimal amount of heat transferred to the object and dT is a resulting
infinitesimal change in temperature. The proportionality constant C is called the heat
capacity of the system.
Equation (2.2-1) does not indicate that C is a derivative of q with respect to T . We
will see that dq is an inexact differential so that the heat capacity C depends on the way
in which the temperature of the system is changed. If the temperature is changed at
constant pressure, the heat capacity is denoted by C P and is called the heat capacity at
constant pressure. If the temperature is changed at constant volume, the heat capacity
is denoted by C V and is called the heat capacity at constant volume. These two heat
capacities are not generally equal to each other.